Noise-induced pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 220

minimum at 0 , the local kinetics

tends to restore this state. For this reason, the deterministic local dynamics are also
called relaxational . The second ingredient is a multiplicative random component that
tends to drive the state of the system away from

φ = φ 0 : If the system is moved away from the state φ

φ = φ 0 . To this end, the multiplicative

function g ( φ ) of the noise term generally has a maximum at φ = φ 0 and decreases as

φ φ

0 . As a result of the balance between deterministic and stochastic
components, bimodal probability distributions of

moves away from

φ

may emerge at steady state. In
fact, when the noise is sufficiently intense the random component is able to maintain
the system far from φ = φ 0 in spite of the fact that the local deterministic dynamics

tend to drive the system toward the neighborhood of φ 0 .

In the case of spatiotemporal dynamical systems, an ordered state may emerge if the
spatial coupling cooperates with the stochastic component to prevent the relaxation
imposed by the local dynamics, therebymaintaining the systemaway from the uniform
state φ = φ 0 . Moreover, the spatial coupling needs to give spatial coherence to the

field, creating a patterned state in which the coherent regions correspond to the two
modes existing in the underlying temporal dynamics. In these conditions, the off-
center (i.e., for φ = φ 0 ) modes of the underlying temporal dynamics are crucial to

the emergence of spatially ordered structures. This mechanism of pattern formation
can then be considered the direct extension to spatially extended systems of noise-
induced transitions in zero-dimensional systems. This mechanism is sometimes called
entropy-driven pattern formation ( Sagues et al. , 2007 ) in that the dynamical system
escapes from the minimum of the potential (i.e., φ = φ 0 ) because of the strength of

noise (which is an entropy source).
It is worth noticing that themechanismof noise-induced pattern formation discussed
in this section has two key differences with respect to those presented in Sections 5.4
and 5.5 , in that (i) it does not emerge from a short-term instability, and (ii) patterns also
emerge if the Langevin equation is interpreted according to Ito's rule. The following
subsection is devoted to presenting this mechanism of pattern formation through a
simple example.

5.7.1 Prototype model
Consider the following spatiotemporal stochastic model:

1
1

∂φ
∂

− D ( k 0 +∇

2 ) 2

t =− a φ +

2 ξ gn ( t )

φ,

(5.73)

+ c φ

where a and c are two positive-valued parameters, ξ gn is a white zero-mean Gaussian

noise with intensity s gn , and the spatial coupling is expressed by the Swift-Hohenberg
operator, where D modulates the coupling intensity and k 0 determines the dominant
wavelength. A similar model was proposed by Buceta et al. ( 2003 ).